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- Jan 26, 2012

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**Problem**: For $x,y>0$, show that

\[\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= 2\int_0^{\pi/2} \cos^{2x-1}\theta\sin^{2y-1}\theta\,d\theta\]

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**Hint**:

\[\Gamma(x) = \int_0^{\infty} e^{-t}t^{x-1}\,dt\]

2) Use (1) to create a double integral for the expression $\Gamma(x)\Gamma(y)$

3) Change to polar coordinates in order to evaluate the double integral.

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